arXiv subjects

Stefan G. Llewellyn Smith

Publications and source records attributed to Stefan G. Llewellyn Smith.

10 recordsLinked to original sources

arXiv2026

Optimal velocity fields for instantaneous magnetic field growth

We consider a variant of the kinematic dynamo problem. Rather than prescribing a velocity field and searching for high-growth magnetic fields via an eigenvalue problem, we treat the seed magnetic-field structure as given and ask which velocity field maximally enhances its instantaneous growth. We show this second problem has an elegant formulation in terms of variational calculus. Upon simultaneously constraining the velocity's kinetic energy and enstrophy, the Euler-Lagrange equation leads to a forced Helmholtz partial differential equation for the optimal velocity field. For the special case of fixed kinetic energy and unconstrained enstrophy, the optimal velocity field everywhere opposes the divergence-free projection of the Lorentz force. In the more general setting, the optimal velocity field can be found through numerical solution of the forced Helmholtz equation. We construct 2.5-dimensional numerical examples to support the theoretical findings, and then leverage the newly found optimal velocity fields to accelerate numerical optimization of the magnetic-velocity field pair for maximal instantaneous growth rate.

physics.flu-dyn↗

arXiv2026

On the fundamental solution for viscous internal waves and Brinkman flows. Part 1. Two dimensions

We obtain the viscous and diffusive fundamental solution for monochromatic internal waves in a uniformly stratified medium and for anisotropic Brinkman flow. These solutions take the form of single integrals with logarithmic singularities, and can be computed numerically in an efficient manner for possible use in boundary integral methods. Far-field asymptotic results are obtained, giving solutions valid far from and inside a ``beam'' corresponding to the internal wave angle in the internal wave case, consistent with Thomas & Stevenson (1972). For Prandtl numbers $\text{Pr} \gtrsim O(1)$, the wave field is given by a superposition of wave- and Stokeslet-like terms. Unlike previous studies, a uniform asymptotic expansion of the wave-field for $\text{Pr} \gtrsim O(1)$ can be computed rigorously. Density diffusion attenuates the wave amplitude as to $(1+\text{Pr}^{-1})^{-2/3}$ and broadens the beam width according to $(1+\text{Pr}^{-1})^{1/3}$. Evanescent waves in a stratified medium and anisotropic Brinkman flows have similar behaviour. Anisotropic Brinkman flow is purely real, dominated by a single circulation cell. As anisotropy increases, the flow becomes increasingly confined to the direction with least resistance. The stratified evanescent wave field has near-vertical cells in its real part, and a dominant single circulation cell in its imaginary part.

physics.flu-dyn↗

arXiv2025

Instability triggered by mixed convection in a thin fluid layer

We investigate the convective stability of a thin, infinite fluid layer with a rectangular cross-section, subject to imposed heat fluxes at the top and bottom and fixed temperature along the vertical sides. The instability threshold depends on the Prandtl number as well as the normalized flux difference ($f$) and decreases with the aspect ratio ($ε$), following a $εf^{-1}$ power law. Using 3D initial value and 2D eigenvalue calculations, we identify a dominant 3D mode characterized by two transverse standing waves attached to the domain edges. We characterize the dominant mode's frequency and transverse wave number as functions of the Rayleigh number and aspect ratio. An analytical asymptotic solution for the base state in the bulk is obtained, valid over most of the domain and increasingly accurate for lower aspect ratios. A local stability analysis, based on the analytical base state, reveals oscillatory transverse instabilities consistent with the global instability characteristics. The source term for this most unstable mode appears to be interactions between vertical shear and horizontal temperature gradients.

physics.flu-dyn↗

arXiv2025

The Unified Transform Method: beyond circular or convex domains

A new transform-based approach is presented that can be used to solve mixed boundary value problems for Laplace's equation in non-convex and other planar domains, specifically the so-called Lipschitz domains. This work complements Crowdy (2015, CMFT, 15, 655--687), where new transform-based techniques were developed for boundary value problems for Laplace's equation in circular domains. The key ingredient of the present method is the exploitation of the properties of the Szegő kernel and its connection with the Cauchy kernel to obtain transform pairs for analytic functions in such domains. Several examples are solved in detail and are numerically implemented to illustrate the application of the new transform pairs.

math.CV↗

arXiv2023

Nusselt number scaling in horizontal convection

We report a numerical study of horizontal convection (HC) at Prandtl number $Pr = 1$, with both no-slip and free-slip boundary conditions. We obtain 2D and 3D solutions and determine the relation between the Rayleigh number $Ra$ and the Nusselt number $Nu$. In 2D we vary $Ra$ between $0$ and $10^{14}$. In the range $10^6 \le Ra \le 10^{10}$ the $Nu$-$Ra$ relation is $Nu \sim Ra^{1/5}$. With $Ra$ greater than about $10^{11}$ we find a 2D regime with $Nu \sim Ra^{1/4}$ over three decades, up to the highest 2D $Ra$. In 3D, with maximum $Ra = 10^{11.5}$, we find only $Nu \sim Ra^{1/5}$. These results apply to both free slip and no slip boundary conditions. The $Nu \sim Ra^{1/4}$ regime has a double boundary layer (BL): there is a thin BL with thickness $\sim Ra^{-1/4}$ nested inside a thicker BL with thickness $\sim Ra^{-1/5}$. The $Ra^{-1/4}$ BL thickness, which determines $Nu$, coincides with the Kolmogorov and Batchelor scales of HC. Numerical and theoretical results indicate that 3D HC is qualitatively and quantitatively similar to 2D HC. At the same $Ra$, the 3D $Nu$ exceeds the 2D $Nu$ by less than $20$%, i.e., there is very little 3D enhancement of heat transport. Boundary conditions are more important than dimensionality: the 2D free-slip solutions have larger $Nu$ than 3D no-slip solutions. Using the mechanical energy power integral of HC we show that the mean square vorticity of 3D HC is nearly equal to that of 2D HC at the same $Ra$. Thus vorticity amplification by strain-mediated vortex stretching does not operate in 3D HC.

physics.flu-dyn↗

arXiv2021

Finite rotating and translating vortex sheets

We consider the rotating and translating equilibria of open finite vortex sheets with endpoints in two-dimensional potential flows. New results are obtained concerning the stability of these equilibrium configurations which complement analogous results known for unbounded, periodic and circular vortex sheets. First, we show that the rotating and translating equilibria of finite vortex sheets are linearly unstable. However, while in the first case unstable perturbations grow exponentially fast in time, the growth of such perturbations in the second case is algebraic. In both cases the growth rates are increasing functions of the wavenumbers of the perturbations. Remarkably, these stability results are obtained entirely with analytical computations. Second, we obtain and analyze equations describing the time evolution of a straight vortex sheet in linear external fields. Third, it is demonstrated that the results concerning the linear stability analysis of the rotating sheet are consistent with the infinite-aspect-ratio limit of the stability results known for Kirchhoff's ellipse (Love 1893; Mitchell & Rossi 2008) and that the solutions we obtained accounting for the presence of external fields are also consistent with the infinite-aspect-ratio limits of the analogous solutions known for vortex patches.

physics.flu-dyn↗

arXiv2020

The Nusselt numbers of horizontal convection

We consider the problem of horizontal convection in which non-uniform buoyancy, $b_{\rm s}(x,y)$, is imposed on the top surface of a container and all other surfaces are insulating. Horizontal convection produces a net horizontal flux of buoyancy, $\mathbf{J}$, defined by vertically and temporally averaging the interior horizontal flux of buoyancy. We show that $\overline{\mathbf{J}\cdot\mathbf{\nabla}b_{\rm s}}=-κ\langle|\boldsymbol{\nabla}b|^2\rangle$; overbar denotes a space-time average over the top surface, angle brackets denote a volume-time average and $κ$ is the molecular diffusivity of buoyancy $b$. This connection between $\mathbf{J}$ and $κ\langle|\boldsymbol{\nabla}b|^2\rangle$ justifies the definition of the horizontal-convective Nusselt number, $Nu$, as the ratio of $κ\langle|\boldsymbol{\nabla}b|^2\rangle$ to the corresponding quantity produced by molecular diffusion alone. We discuss the advantages of this definition of $Nu$ over other definitions of horizontal-convective Nusselt number currently in use. We investigate transient effects and show that $κ\langle|\boldsymbol{\nabla}b|^2\rangle$ equilibrates more rapidly than other global averages, such as the domain averaged kinetic energy and bottom buoyancy. We show that $κ\langle|\boldsymbol{\nabla} b|^2\rangle$ is essentially the volume-averaged rate of Boussinesq entropy production within the enclosure. In statistical steady state, the interior entropy production is balanced by a flux of entropy through the top surface. This leads to an equivalent "surface Nusselt number", defined as the surface average of vertical buoyancy flux through the top surface times the imposed surface buoyancy $b_{\rm s}(x,y)$. In experiments it is likely easier to evaluate the surface entropy flux, rather than the volume integral of $|\mathbf{\nabla}b|^2$ demanded by $κ\langle|\mathbf{\nabla}b|^2\rangle$.

physics.flu-dyn↗

arXiv2019

Numerical solution of scattering problems using a Riemann--Hilbert formulation

A fast and accurate numerical method for the solution of scalar and matrix Wiener--Hopf problems is presented. The Wiener--Hopf problems are formulated as Riemann--Hilbert problems on the real line, and a numerical approach developed for these problems is used. It is shown that the known far-field behaviour of the solutions can be exploited to construct numerical schemes providing spectrally accurate results. A number of scalar and matrix Wiener--Hopf problems that generalize the classical Sommerfeld problem of diffraction of plane waves by a semi-infinite plane are solved using the approach.

math.NA↗

arXiv2014

Short- and Long- Time Transport Structures in a Three Dimensional Time Dependent Flow

Lagrangian transport structures for three-dimensional and time-dependent fluid flows are of great interest in numerous applications, particularly for geophysical or oceanic flows. In such flows, chaotic transport and mixing can play important environmental and ecological roles, for examples in pollution spills or plankton migration. In such flows, where simulations or observations are typically available only over a short time, understanding the difference between short-time and long-time transport structures is critical. In this paper, we use a set of classical (i.e. Poincaré section, Lyapunov exponent) and alternative (i.e. finite time Lyapunov exponent, Lagrangian coherent structures) tools from dynamical systems theory that analyze chaotic transport both qualitatively and quantitatively. With this set of tools we are able to reveal, identify and highlight differences between short- and long-time transport structures inside a flow composed of a primary horizontal contra-rotating vortex chain, small lateral oscillations and a weak Ekman pumping. The difference is mainly the existence of regular or extremely slowly developing chaotic regions that are only present at short time.

physics.flu-dyn↗

arXiv1999

Radiation of mixed layer near-inertial oscillations into the ocean interior

The radiation from the mixed layer into the interior of the ocean of near-inertial oscillations excited by a passing storm in the presence of the beta effect is reconsidered as an initial-value problem. Making use of the fact that the mixed layer depth is much smaller than the total depth of the ocean, the solution is obtained in the limit of an ocean that is effectively infinitely deep. For a uniform initial condition, analytical results for the velocity, horizontal kinetic energy density and fluxes are obtained. The resulting decay of near-inertial mixed layer energy in the presence of the beta effect occurs on a timescale similar to that observed.

physics.ao-ph↗